Question

The total cost in dollars of producing x lawn mowers is given by C(x) = 4,000 + 90x - x^2/3. What does the marginal average cost at x = 20, C'(20), represent? A) The total cost of producing 20 lawn mowers B) The change in total cost when producing 20 more lawn mowers C) The average cost of producing 20 lawn mowers D) The additional cost of producing one more lawn mower at x = 20

Answer 1

C) The average cost of producing 20 lawn mowers.

Step-by-step explanation:

The marginal average cost at
`\( x = 20 \),` denoted as
`\( C'(20) \)`, represents the rate of change of the average cost with respect to the number of lawn mowers produced. Mathematically,
`\( C'(x) \)` is the derivative of the total cost function
`\( C(x) \) with respect to \( x \). To find \( C'(20) \), we calculate the derivative of \( C(x) = 4,000 + 90x - (x^2)/(3) \) and evaluate it at \( x = 20 \).`

`\[ C'(x) = 90 - (2x)/(3) \]`

Substitute
`\( x = 20 \) into \( C'(x) \):`

`\[ C'(20) = 90 - (2 * 20)/(3) = 90 - (40)/(3) = (270 - 40)/(3) = (230)/(3) \]`

So,
`\( C'(20) \)`is the rate of change of the average cost at
`\( x = 20 \).` This represents how the average cost changes when producing one more lawn mower at that specific production level.

Understanding the concept of marginal average cost is crucial in economics and business. It provides insights into the additional cost incurred for each additional unit produced, helping businesses make informed decisions about production levels and pricing strategies. In this context,
`\( C'(20) \)` specifically tells us about the average cost change at the production quantity of 20 lawn mowers.